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Erschienen in: Applicable Algebra in Engineering, Communication and Computing 4/2022

19.09.2020 | Original Paper

InfoMod: a visual and computational approach to Gauss’ binary quadratic forms

verfasst von: Ayberk Zeytin

Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 4/2022

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Abstract

InfoMod is a software devoted to the modular group, \(\mathrm {PSL}_2 (\mathbf {Z})\). It consists of algorithms that deal with the classical correspondences among geodesics on the modular surface, elements of the modular group and binary quadratic forms. In addition, the software implements the recently discovered representation of Gauss’ indefinite binary quadratic forms and their classes in terms of certain infinite planar graphs (dessins) called çarks. InfoMod illustrates various aspects of these forms, i.e. Gauss’ reduction algorithm, the representation problem of forms, ambiguous and reciprocal forms. It can be used for visualization, for high performance computation involving these mathematical structures as well as experimenting.

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Fußnoten
1
The word çark (pronounced as chark ) is a Turkish word, borrowed from Persian. It has a common etymology with Indian chakra, Greek kyklos and English wheel.
 
2
One may define attracting and repelling fixed points of W by looking at its action along this geodesic. This distinction is unnecessary for our purposes.
 
3
The length of a word is defined to be the number of letters (L, \(L^{2}\) and S) appearing in the word.
 
4
A bipartite graph is a graph whose vertices can be decomposed into two disjoint sets so that no two edge belonging to the same set are joined by an edge. A ribbon graph is a graph together with a cyclic ordering of edges emanating from each vertex.
 
5
Standard ActionScript libraries do provide a native \(3 \times 3\) matrix class, but in our case we needed some extra functions like trace,transpose and multiplication with specific constants (i.e. generators of modular group and some of their compositions) which reduces to simple swap and sign change of values when optimized.
 
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Metadaten
Titel
InfoMod: a visual and computational approach to Gauss’ binary quadratic forms
verfasst von
Ayberk Zeytin
Publikationsdatum
19.09.2020
Verlag
Springer Berlin Heidelberg
Erschienen in
Applicable Algebra in Engineering, Communication and Computing / Ausgabe 4/2022
Print ISSN: 0938-1279
Elektronische ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00460-w

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